Summary: In this paper, we show that if a rectifiable space \(G\) is a locally Lindelöf \(\sum \)-space, then \(G\) is strongly paracompact. We prove that if a rectifiable space \(G\) has a compactification \(bG\) such that \(bG\backslash G\) is a locally \(\sigma \)-space, then \(G\) is either locally compact or separable and metrizable. We also show that if a non-locally compact \(k\)-gentle paratopological group \(G\) has a compactification \(bG\) such that \(bG\backslash G\) has locally a \({G_\delta}\)-diagonal, then either \(G\) is a \(\sigma \)-compact cosmic space, or \(bG\) is separable and metrizable.